Realisability of $G_{n}^{3}$, realisability projection, and kernel of the $G_{n}^{3}$-braid presentation

The aim of this article is to prove that the kernel of the map from the pure braid group PB n , n ≥ 4 to the group G 3 n consists of full twist braids and their exponents. The proof consists of two parts. The first part which deals with n = 4 relies on the crucial tool in this construction having its own interest is the realisability projection saying that if two realisable G 34 -elements are equivalent then they are equivalent by a sequence of realisable ones. The second part (an easy one) uses induction on n .